10.8 Fourier-Bessel series
So how can we determine in general the
coefficients in the Fourier-Bessel series
|
(10.71) |
The corresponding self-adjoint version of
Bessel’s equation is easily found to be (with
)
|
(10.72) |
Where we assume that
and
satisfy the boundary condition
From Sturm-Liouville theory we do know
that
|
(10.74) |
but we shall also need the values when
!
Let us use the self-adjoint form of the
equation, and multiply with
, and integrate over
from
to
,
|
(10.75) |
This can be brought to the form (integrate
the first term by parts, bring the other two terms to the
right-hand side)
The last integral can now be done by
parts:
So we finally conclude that
|
(10.79) |
In order to make life not too complicated we
shall only look at boundary conditions where
. The other cases (mixed or purely
) go very similar! Using the fact that
, we find
|
(10.80) |
We conclude that
We thus finally have the result
|
(10.82) |
Example
10.1:
-
Consider the function
|
(10.83) |
Expand this function in a
Fourier-Bessel series using
.
Solution:
-
From our definitions we
find that
|
(10.84) |
with
Using
, we find that the first five
values of
are
. The first five partial sums are
plotted in Fig.
10.6
.